Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond. D oron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book. To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses.…